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1. Bisection Method Major: All Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates http://numericalmethods.eng.usf.edu

2. Bisection Methodhttp://numericalmethods.eng.usf.edu

3. Basis of Bisection Method Theorem An equation f(x)=0, where f(x) is a real continuous function, has at least one root between xl and xu if f(xl) f(xu) < 0. Figure 1 At least one root exists between the two points if the function is real, continuous, and changes sign. http://numericalmethods.eng.usf.edu

4. Basis of Bisection Method Figure 2 If function does not change sign between two points, roots of the equation may still exist between the two points. http://numericalmethods.eng.usf.edu

5. Basis of Bisection Method Figure 3 If the function does not change sign between two points, there may not be any roots for the equation between the two points. http://numericalmethods.eng.usf.edu

6. Basis of Bisection Method Figure 4 If the function changes sign between two points, more than one root for the equation may exist between the two points. http://numericalmethods.eng.usf.edu

7. Algorithm for Bisection Method http://numericalmethods.eng.usf.edu

8. Step 1 Choose xl and xu as two guesses for the root such that f(xl) f(xu) < 0, or in other words, f(x) changes sign between xl and xu. This was demonstrated in Figure 1. Figure 1 http://numericalmethods.eng.usf.edu

9. Step 2 Estimate the root, xm of the equation f (x) = 0 as the mid point between xl and xu as Figure 5 Estimate of xm http://numericalmethods.eng.usf.edu

10. Step 3 Now check the following • If , then the root lies between xl and xm; then xl = xl ; xu = xm. • If , then the root lies between xm and xu; then xl = xm; xu = xu. • If ; then the root is xm. Stop the algorithm if this is true. http://numericalmethods.eng.usf.edu

11. Step 4 Find the new estimate of the root Find the absolute relative approximate error where http://numericalmethods.eng.usf.edu

12. Step 5 Compare the absolute relative approximate error with the pre-specified error tolerance . Go to Step 2 using new upper and lower guesses. Yes Is ? No Stop the algorithm Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it. http://numericalmethods.eng.usf.edu

13. Example 1 You are working for ‘DOWN THE TOILET COMPANY’ that makes floats for ABC commodes. The floating ball has a specific gravity of 0.6 and has a radius of 5.5 cm. You are asked to find the depth to which the ball is submerged when floating in water. Figure 6 Diagram of the floating ball http://numericalmethods.eng.usf.edu

14. Example 1 Cont. The equation that gives the depth x to which the ball is submerged under water is given by a) Use the bisection method of finding roots of equations to find the depth x to which the ball is submerged under water. Conduct three iterations to estimate the root of the above equation. b) Find the absolute relative approximate error at the end of each iteration, and the number of significant digits at least correct at the end of each iteration. http://numericalmethods.eng.usf.edu

15. Example 1 Cont. From the physics of the problem, the ball would be submerged between x = 0 and x = 2R, where R = radius of the ball, that is Figure 6 Diagram of the floating ball http://numericalmethods.eng.usf.edu

16. Example 1 Cont. Solution To aid in the understanding of how this method works to find the root of an equation, the graph of f(x) is shown to the right, where Figure 7 Graph of the function f(x) http://numericalmethods.eng.usf.edu

17. Example 1 Cont. Let us assume Check if the function changes sign between xl and xu . Hence So there is at least on root between xl and xu, that is between 0 and 0.11 http://numericalmethods.eng.usf.edu

18. Example 1 Cont. Figure 8 Graph demonstrating sign change between initial limits http://numericalmethods.eng.usf.edu

19. Example 1 Cont. Iteration 1 The estimate of the root is Hence the root is bracketed between xm and xu, that is, between 0.055 and 0.11. So, the lower and upper limits of the new bracket are At this point, the absolute relative approximate error cannot be calculated as we do not have a previous approximation. http://numericalmethods.eng.usf.edu

20. Example 1 Cont. Figure 9 Estimate of the root for Iteration 1 http://numericalmethods.eng.usf.edu

21. Example 1 Cont. Iteration 2 The estimate of the root is Hence the root is bracketed between xl and xm, that is, between 0.055 and 0.0825. So, the lower and upper limits of the new bracket are http://numericalmethods.eng.usf.edu

22. Example 1 Cont. Figure 10 Estimate of the root for Iteration 2 http://numericalmethods.eng.usf.edu

23. Example 1 Cont. The absolute relative approximate error at the end of Iteration 2 is None of the significant digits are at least correct in the estimate root of xm = 0.0825 because the absolute relative approximate error is greater than 5%. http://numericalmethods.eng.usf.edu

24. Example 1 Cont. Iteration 3 The estimate of the root is Hence the root is bracketed between xl and xm, that is, between 0.055 and 0.06875. So, the lower and upper limits of the new bracket are http://numericalmethods.eng.usf.edu

25. Example 1 Cont. Figure 11 Estimate of the root for Iteration 3 http://numericalmethods.eng.usf.edu

26. Example 1 Cont. The absolute relative approximate error at the end of Iteration 3 is Still none of the significant digits are at least correct in the estimated root of the equation as the absolute relative approximate error is greater than 5%. Seven more iterations were conducted and these iterations are shown in Table 1. http://numericalmethods.eng.usf.edu

27. Table 1 Cont. Table 1 Root of f(x)=0 as function of number of iterations for bisection method. http://numericalmethods.eng.usf.edu

28. Table 1 Cont. Hence the number of significant digits at least correct is given by the largest value or m for which So The number of significant digits at least correct in the estimated root of 0.06241 at the end of the 10th iteration is 2. http://numericalmethods.eng.usf.edu

29. Advantages • Always convergent • The root bracket gets halved with each iteration - guaranteed. http://numericalmethods.eng.usf.edu

30. Drawbacks • Slow convergence • If one of the initial guesses is close to the root, the convergence is slower http://numericalmethods.eng.usf.edu

31. Drawbacks (continued) • If a function f(x) is such that it just touches the x-axis it will be unable to find the lower and upper guesses. http://numericalmethods.eng.usf.edu

32. Drawbacks (continued) • Function changes sign but root does not exist http://numericalmethods.eng.usf.edu

33. Additional Resources For all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit http://numericalmethods.eng.usf.edu/topics/bisection_method.html

34. THE END http://numericalmethods.eng.usf.edu